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The Mixing Time of the Newman-Watts Small-World Model

Part of: Graph theory Combinatorial probability

Published online by Cambridge University Press:  04 January 2016

Louigi Addario-Berry  and
Tao Lei
Louigi Addario-Berry*
Affiliation:
McGill University
Tao Lei*
Affiliation:
McGill University
*
∗ Postal address: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montréal, Québec, H3A 0B9, Canada.
∗ Postal address: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montréal, Québec, H3A 0B9, Canada.
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Abstract

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‘Small worlds’ are large systems in which any given node has only a few connections to other points, but possessing the property that all pairs of points are connected by a short path, typically logarithmic in the number of nodes. The use of random walks for sampling a uniform element from a large state space is by now a classical technique; to prove that such a technique works for a given network, a bound on the mixing time is required. However, little detailed information is known about the behaviour of random walks on small-world networks, though many predictions can be found in the physics literature. The principal contribution of this paper is to show that for a famous small-world random graph model known as the Newman-Watts small-world model, the mixing time is of order log2n. This confirms a prediction of Richard Durrett [5, page 22], who proved a lower bound of order log2n and an upper bound of order log3n.

Keywords

Random graphsmall worldmixing timeconductance bound

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C81: Random walks on graphs 05C82: Small world graphs, complex networks
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 1 , March 2015 , pp. 37 - 56
Copyright
© Applied Probability Trust 

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